 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cleljust Structured version   Visualization version   GIF version

Theorem cleljust 2115
 Description: When the class variables in definition df-clel 2774 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2108 with the class variables in wcel 2107. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2052 in order to remove dependencies on ax-10 2135, ax-12 2163, ax-13 2334. Note that there is no disjoint variable condition on 𝑥, 𝑦, that is, on the variables of the left-hand side, as should be the case for definitions. (Revised by BJ, 29-Dec-2020.)
Assertion
Ref Expression
cleljust (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljust
StepHypRef Expression
1 elequ1 2114 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
21equsexvw 2052 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
32bicomi 216 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386  ∃wex 1823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824 This theorem is referenced by:  bj-dfclel  33460
 Copyright terms: Public domain W3C validator